In short, by 1961, when Tiffany's was released, the Hay's Code was no longer taken seriously. Remember that the Code – MPAA Production Code, to use its proper name – was a set of guidelines for self-censorship mutually agreed upon by the major Hollywood studios (and followed of necessity by the minors as well). It was never law; in fact, its whole purpose was to preempt centralized federal censorship in the U.S. When the studios decided to stop following those guidelines, well, there was no particular repercussion.

So why did they do so? Basically, large shifts in American society and social mores between the 1930s and the 1950s. This means on the one hand a weakening of conservative forces like the (Catholic) National League of Decency, which was instrumental in getting the Code passed in the first place, and on the other hand, a greater tolerance towards, and of course interest in, adult subject matter being explored in the cinemas. One might also add the economic pinch that studios were feeling in the wake of the U.S. vs. Paramount decision in 1948 (which stripped them of their exhibition business) and the ruinous competition that the television proved in the postwar suburbanizing world. They were willing to try anything.

On the social side of things, we might point to any number of key points in the erosion of public support for the Code. The most salient, I think, is the experience of WWII, and the generation of adults who emerged in its wake. They were much less willing to accept a form of mass entertainment unwilling and unable to showcase anything of the messy human reality of adult life. For many, it was the influx of European (and eventually Japanese) films that were "real" in a way Hollywood movies never had been, while also showcasing a staggering level of artistry. Did you know that Bicycle Thieves was originally banned in the US? Not (ostensibly) for any political content – but because there's a brief scene when the son, with his back to the camera, starts to pee against a wall. Moviegoers were incensed. Here was a lauded, prizewinning picture from the most exciting film country on earth...and the MPAA was saying a boy peeing on a wall was too much for delicate American sensibilities? It made them a bit of a laughingstock.

This would become something of a theme into the 1950s, especially with the rise of powerful producer-directors like Otto Preminger and Billy Wilder. Films by these two (among others) routinely flouted the MPAA guidelines with controversial subject matter, and though the MPAA pushed back, controversy is of course the best publicity. Though the code would lumber on zombie-like until 1968, 1959's Some Like It Hot is probably the real nail in its coffin as a meaningful regulatory body. That film was produced entirely outside the MPAA's approval, it made a ton of money, and was a hit with the critics too.

I quite recommend The Dame in the Kimono for an accessible history of censorship in Hollywood.